Here is the full list of all 20 Pony Island achievements. It takes around 4-5 hours to unlock all of the achievements on Windows.
Destroyed the third Core File.
Found all 24 tickets.
Killed Louey.
Learned a third secret of your past.
Destroyed the second Core File.
Found Missingno.
Learned another secret of your past.
Found an alternate avatar.
Found 14 tickets.
Unlocked Pony Wings.
Found the hidden profile screen.
Destroyed the first Core File.
Learned a secret of your past.
Found 7 tickets.
Crashed the game.
Found 3 tickets.
Found your first ticket.
Unlocked Pony Lasers.
Escaped the arcade.
Learned every secret of your past.